Make a chord with slope k (K ≠ 0) through the focus of parabola y ^ 2 = 4x The chord meets the following requirements: 1. The chord length does not exceed 8; 2. The straight line where the chord is located intersects the ellipse 3x ^ 2 + 2Y ^ 2 = 2, and the value range of K is calculated,

Make a chord with slope k (K ≠ 0) through the focus of parabola y ^ 2 = 4x The chord meets the following requirements: 1. The chord length does not exceed 8; 2. The straight line where the chord is located intersects the ellipse 3x ^ 2 + 2Y ^ 2 = 2, and the value range of K is calculated,

1. Let y = K (x-1)
2. Substituting the parabola, we can get X1 + X2, X1 * X2, with / AB / = √ K ^ 2 + 1 * √ (x1 + x2) ≤ 8
Calculate K ^ 2 > = 1
3. Generation ellipse, calculate △ = 24-8k ^ 2 > = 0, can get k ^ 2

The midpoint of chord PQ of parabola y ^ 2 = 2px (P > 0) is m (x0, Y0) (Y0 ≠ 0) to find the slope of straight line PQ

Let P coordinate be (x1, Y1) and Q coordinate be (x1, Y2), then 2yo = Y1 + Y2
Y1^2=2px1
y2^2=2px2
By subtracting the two formulas, we get the following result:
(y1+y2)(y1-y2)=2p(x1-x2)
That is, PQ slope k = (y1-y2) / (x1-x2) = 2P / (Y1 + Y2) = 2P / 2yo = P / yo

If the midpoint of the chord PQ of the parabola y ^ 2 = 2px is m (x0, Y0) (Y0 does not = 0), the slope of the straight line PQ is obtained

Let the midpoint of a (x0, Y0) and B (x1, Y1) be m. by subtracting the parabola of generation a and B, we get: Y1 & sup2; - Y2 & sup2; = 2p (x1-x2), (y1-y2) / (x1-x2) = 2P / (Y1 + Y2) = 2P / (2y0) = P / Y0, that is, k = P / (Y0)

It is known that the parabola y ^ 2 = 2px, P (x0, Y0) line L passes through point P and intersects with the parabola at two points a and B. if the chord AB is exactly bisected by point P, it is proved that the slope of line L is p / Y0

A(X1,Y1) B(X2,Y2)
So X1 + x2 = 2x0
Y1+Y2=2Y0
Let K be the slope of the line
Y-Y0=K(X-XO)
A. B two points are on a straight line
Y1-Y0=K(X1-X0)
Y2-Y0=K(X2-X0)
Want to subtract y1-y2 = K (x1-x2)
Y1^2=2PX1
Y2^2=2PX2
Subtracting (y1-y2) (Y1 + Y2) = 2p (x1-x2)
(Y1-Y2)/(X1-X2) * 2YO=2P
K* 2YO=2P
K=P/Y0